Kurt Warner announced his retirement Friday, and his Hall of Fame credentials have been a topic of conversation. Everyone is familiar with his career: three Super Bowl appearances, one ring, two MVP awards, and a compelling personal story that inspires grocery stock clerks around the country.

According to Pro-Football-Reference, his career Adjusted Net Yards Per Pass Attempt (net passing efficiency minus 45 yards for each interception and with a 20-yard bonus for each TD pass) is 6.7 ANY/A. That's very, very good, and it puts him in league with the era's best passers. But it's not called the Hall of Efficiency; it's the Hall of Fame.

With that in mind, we turn to Win Probability Added (WPA), a narrative stat uniquely suited for measuring performance in terms of how each play helps a team win or lose. Fame is about a lot of things, but it's primarily about wins, so let's see what WPA has to say.

This post will explain the concepts of Expected Points and Expected Points Added. In future posts when I refer to these stats, I'll link here.

Football is a sport of strategy and decision making. But before we can compare the potential risks and rewards of various options, we need to be able to properly measure the value of possible outcomes.

The value of a football play has traditionally been measured in yards gained. Unfortunately, yards is a flawed measure because not all yards are equal. For example, a 4-yard gain on 3rd down and 3 is much more valuable than a 4-yard gain on 3rd and 8. Any measure of success must consider the down and distance situation.

Field position is also an important consideration. Yards gained near the goal line are tougher to come by and are more valuable than yards gained at midfield. Yards lost near one’s own goal line can be more costly as well.

Super Bowl game probabilities are available now at the nytimes.com Fifth Down. This week I also review the model's accuracy this season and discuss the how the concept of hindsight bias misleads us about our ability to predict future events.

I’ve begun using some new stats here lately, and I want to take the opportunity to explain them clearly. Readership has been increasing rapidly too, and some readers aren't as familiar with the terms we use around here. This post will serve as a reference each time I use the new stats in the future.

Although it’s been used in baseball sabermetrics for several years now, Win Probability Added (WPA) is a new, or at least rediscovered, concept for football stats. It measures each play in terms of how much it increased or decreased a team’s chances of winning the game.

WPA starts with a Win Probability (WP) model of the game of football. Every situation in a game gives each opponent a particular chance of winning, and a WP model estimates those chances. The model created here at Advanced NFL Stats uses score, time, down, distance, and field position to estimate how likely each team will go on to win the game. For example, at the start of the 2nd quarter, a team down by 7 points with a 2nd down and 5 from their own 25 will win about 36% of the time--in other words a 0.36 WP.

There's a lot to cover in this one. It was the perfect Sunday to unveil the real-time comment section on the live WP graph pages. Unfortunately, it was too perfect. The server was overloaded with an explosion in traffic, and I don't think many people were able to see the graphs.

Let's go in reverse chronological order and start with the overtime. The NFL overtime rules are far too arbitrary. A coin flip gave one team the ball and the other never had a chance to respond. This happens in one third of all overtimes, and this time it happened in one of the biggest, most exciting games in recent memory. You can say, "Well, the Vikings had a chance to make a stop." True. But the Saints defense never had to. Because the Saints managed a field goal, their defense was never even asked to take the field. Their "stop" was automatic. There's a lot more wrong with the way NFL overtime works. You can read about it here (or here).

I'm testing out a great suggestion a reader made a couple weeks ago. (Sorry, can't remember who!) The live Win Probability Graphs now feature comments. You can put in your two cents on coaching decisions, remarkable plays, or ask general questions about how the graphs work. Help me test it out during today's conference championship games.

A lot of analysis of running vs. passing takes into account the added risk of interceptions. Almost all sources of passing stats will include team or player interceptions. But fumbles are more tricky. Stat sites will tell you team fumbles, but they usually won't tell you how many were due to rushes or due to passes.

Unlike interceptions, fumbles can happen on either type of play. But is the risk of a fumble even between runs and passes, or are fumbles more likely to occur on one or the other type of play? Further, what about fumbles lost? Are fumbles more likely to be lost on runs or passes? And how does the sack-fumble factor in?

I’ve been writing a lot about run-pass balance lately, and part of my theory of why teams are perhaps passing less often than they should has to do with the evolution of the sport. Rule changes over the recent decades have generally favored passing. Changes in pass blocking rules and in pass interference rules have made it easier to pass the ball successfully. Even subtle rule changes such as the definition of possession and “control” may have made receiver fumbles less likely.

Tactics and play selection have been refined over the years to take advantage of the rule changes, but I’m not sure that they’ve completely caught up. Results from several studies, including my own, have suggested that in most situations, passing is more lucrative than running. This imbalance implies that passing should be selected more often. As defenses respond to expect more frequent passing, the payoff for passes will decrease as the payoff for runs increases. Eventually, there is an equilibrium where the payoffs should be equal.

In this post, I’m going to look at very simple historical trends. As you’ll see in the graphs below, there is evidence that the current run-pass balance has not responded fully to recent increases in the payoff of passes. All data come from PFR's very cool league historical pages.

Just before halftime in last year's Super Bowl, on first and goal from the one, Kurt Warner threw the ball directly into the arms of James Harrison who rumbled 100 yards for a touchdown. With so little time left in the half, passing was the obvious call, but that play highlights the dangers of passing so close to the goal line.

Game theory tells us that when payoffs for strategies are unequal, the strategy with the higher payoff should be chosen more often. We've seen that between the 20 yard lines payoffs for passes are consistently higher than for runs on 1st down, but inside the 20 running becomes more lucrative. Now let's take a look at the red zone in more detail, where the stakes get higher and the field gets shorter. On 1st downs in the red zone, should offenses run or pass more often, or do they already strike the right balance?

With just over 2 minutes to go and only 1 timeout, Chargers head coach Norv Turner faced a tough call. After scoring a touchdown to cut the deficit to 3 points, he could attempt on onside kick, or he could kick away. Turner elected to go for the onside kick. Was it a good decision? What about Rex Ryan's decision to go for it on 4th down to seal the win?

Onside kicks, when expected, are successful just 20% of the time. A successful kick would have meant a first down at the Chargers' own 40 or so. This equates to a win probability (WP) of 0.26. A failed onside attempt gives the Jets a first down at the same field position, giving the Chargers a 0.07 WP. On net, an onside attempt is worth:

Last week we saw the Ravens stomp the Patriots throwing only 10 times for 34 yards. Chase Stuart looks at the reverse. How close have NFL teams come to never running the ball in a game? Could we see something like that in today's Cardinals-Saints game?

Indianapolis, New Orleans, and San Diego are still the big three headed into the divisional round games. My system is not very high on Minnesota and Arizona, and it could very well be wrong about those teams. But it's worth looking at why they don't fare well in the projections.

Let's look at the efficiency stats as of week 16 for the teams still alive. (The projections below include performance from the wildcard games but exclude week 17.) We'll see why the projections shake out the way they do.

Or The Seasons That Weren't, perhaps. Chris from NFL-Forecast.com reran the 2009 season schedule in a Monte Carlo simulation to see how different the actual season turned out from the most likely outcomes.

The NFC ends up pretty much as you'd expect. In the AFC, the big surprise is Pittsburgh, who according to their stats would expect to make the playoffs 72% of the time.

The Packers' season came to an abrupt end against the Cardinals, but head coach Mike McCarthy's unorthodox tactics were not the cause. McCarthy engineered an improbable comeback that sent the game into overtime

Following a Packers touchdown that brought the score to within 14 points in the third quarter, McCarthy called for an onside kick. Onside kicks are surprisingly successful when unexpected, averaging a 60% successful recovery rate. In this instance, it worked, and the Packers were on their way to another touchdown drive. And just as important, it kept the ball out of the hands of the potent Cardinals offense.

Here is a handy comparison of the efficiency stats for the current playoff teams. The past couple years, I found myself referring to this chart myself many times throughout the playoffs to get a feel for how teams might match up.

O Pass, D Run, etc. is in yards per attempt. O Int, and D Int are the percent of pass attempts resulting in an interception. O Fum Rate is fumbles per offensive run play (all fumbles, not just fumbles lost). Pen rate is penalty yards per all plays. These are the stats most predictive of game outcomes.

Indianapolis is your Super Bowl favorite, if only by a couple of percent. New Orleans is right behind, and San Diego rounds out the top three. After those teams, it looks like table scraps, either due to team strength or lack of home field advantage, or both. My money says this season's champion is going to be one of those three teams. Then again, anything can happen. The 2007 Giants and the 2008 Cardinals (almost) proved that.

Courtesy of NFL-Forecast.com, here are your playoff probabilities going into the wildcard games.

It's a little silly to consider these the final rankings, particularly for the top teams. After week 17's meaningless or almost-meaningless games, many of which were blowouts, it makes more sense to consider last week's rankings as truer estimates of team strength. Even for the teams that played full strength games, many of their previous opponents did not, which affects their strength of schedule and ultimately, their own final rankings. For the record, however, here are the 2009 regular season team efficiency rankings.

I've been waiting for a good opportunity to write this article for a while. Recently, Carl Bialik at the WSJ asked me to take a look at some punting stats for a post on the 'Golden Age of Punting.' For the last few years, Shane Lechler has been lauded for his super-human punting ability. For three of the last six years he's led the league in total punting yards, and for five of the last seven years he's lead the league in yards per punt. He'll be going to his fourth pro-bowl in a few weeks.

But, there's a problem with all those gaudy stats. Lechler has the benefit of playing for the Oakland Raiders, who for the last few seasons have fielded a terrible offense. Lechler, more than most other punters, gets to punt from deep in his own territory where the chance of a touchback doesn't shorten punts. We can call this the JaMarcus Effect.

In this post, I'll look at where Lechler's average field position is compared to the rest of the league. We'll see that this has a big effect on punt distances. But we'll also see that I'm wrong about Lechler in the end. Despite this unfair advantage, he's still the NFL's best.

Going into tonight's Bengals-Jets game, the playoff situation is pretty simple. Baltimore is in after their victory at Oakland, which also eliminates Pittsburgh. Denver is out following their loss at Kansas City. Only the Jets and Texans are still alive.

Assuming full-strength play, I'd have the Jets as 63% favorites, making the Texans' chances 37%. We'll see how hard Cincinnati plays.

The past year was a very good one for this website in several ways. Certainly if the visitor traffic is any indication, interest in advanced football analysis is growing by leaps and bounds. And I hope most readers would agree that this site has produced more high-quality articles in 2009 than ever.

Judging from the emails I get, Advanced NFL Stats fills a niche that's either been neglected or poorly served for years. Your encouraging notes are a part of the reason I continue to invest time in the site. The best compliments are the ones I come across on message boards or other sites because they're the most genuine. I had been keeping a list of all the best comments until I saw one that topped them all. It said something like, "best football site ever...my only criticism is that they don't post more often."

The site's readership has more than quadrupled over the course of the current season, and it continues to grow. I realize the site has the feel of a small amateur blog, which it is. But believe it or not, the readership here is higher than for most magazines you see on the rack in the bookstore. I hope many of the new readers have gone back through the archives to read some of the nuggets from months and years past. But I know it's hard to spend time combing through hundreds of posts looking for the best ones. (Even though I can tell some of you do judging by the visitor logs!)

So in honor of the new year, here's a list of what I think are the top articles from 2009. I chose them based on either the attention they received or, frankly, my own opinion. I'll go in chronological order:

@BBurkeESPN

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